Order reduction for nonlinear dynamic models of distributed reacting systems

Detailed first-principles models of transport and reaction (based on partial differential equations) lead, after discretization, to dynamical systems of very high order. Systematic methodologies for model order reduction are vital in exploiting such fundamental models in the analysis, design and real-time control of distributed reacting systems. We briefly review some approaches to model order reduction we have successfully used in recent years, and illustrate their capabilities through (a) the design of an observer and stabilizing controller of a reaction-diffusion problem and (b) two-dimensional simulations of the transient behavior of a horizontal MOVPE reactor.     

A practical approach for hybrid distributed MPC

This paper presents a framework to deal with distributed optimization problems composed by binary and continuous variables. Instead of using a mixed integer quadratic programming (MIQP), the approach proposed here transforms the MIQP into a set of quadratic programming's (QP) that are easier to solve. In this way an instance of the controller related to each feasible combination of binary variables is created. The distributed controller performs an iterative process where the set of agents must agree on the value of continuous interconnection variables, while each agent must decide the values of local binary variables. During the iteration procedure the instances are rated according to a performance index and the instances with best performance are selected until the best one is obtained. The proposed methodology is applied to economic optimization of networked microgrids.     

基于特征线法的双曲型分布参数模型预测控制及其应用

本文针对双曲型分布参数系统提出基于特征线法的模型预测控制算法. 通过特征线变换将描述分布参数模型的偏微分方程转化为常微分方程; 进而求解得到分布参数系统状态变量的解析式; 离散化后作为预测模型用于模型预测控制. 以循环流化床烟气脱硫系统中SO2浓度控制为例, 进行仿真研究, 结果表明基于特征线法的模型预测控制算法可以实现对双曲型分布参数系统的有效控制, 并且该算法的控制效果优于目前工程应用的前馈反馈 控制策略.     

A self-tuning regulator for distributed parameter systems

The control of distributed parameter systems with constant, but unknown parameters is considered. A weighted average of the distributed output on the spatial domain is defined as a new variable and is used to generate the control. The parameters of the model are estimated using recursive least squares estimation. The control is obtained using a minimum variance strategy based on the estimated parameters. Distributed disturbances and measurement noise are allowed to be present. Measurements at a finite number of points in the spatial domain are used in obtaining a discrete-time model. From the simulation of a one-sided heating diffusion process the self-tuning regulator is shown to have attractive characteristics and hence can be recommended for practical on-line control of distributed parameter systems.     

Observer-based event-triggered distributed model predictive control for a class of nonlinear interconnected systems

In this paper, an observer-based event-triggered distributed model predictive control method is proposed for a class of nonlinear interconnected systems with bounded disturbances, considering unmeasurable states. First of all, the state observer is constructed. It is proved that the observation error is bounded. Second, distributed model predictive controller is designed by using observed value. Meanwhile, the event-triggered mechanism is set by using the error between the actual output and the predicted output. The setting of event-triggered mechanism not only ensures the error between the actual output and the predicted output within a certain range, but also reduces the calculation amounts of solving the optimization problem. The states of each subsystem enter the terminal invariant set by distributed model predictive control, and then are stabilized in the invariant set under the action of output feedback control law. In addition, sufficient conditions are given to ensure the feasibility of the algorithm and the stability of the closed-loop system. Finally, the numerical example is given, and the simulation results verify the effectiveness of the proposed algorithm.     

A robust multi-model predictive controller for distributed parameter systems

In this work a robust nonlinear model predictive controller for nonlinear convection-diffusion-reaction systems is presented. The controller makes use of a collection of reduced order approximations of the plant (models) reconstructed on-line by projection methods on proper orthogonal decomposition (POD) basis functions. The model selection and model update step is based on a sufficient condition that determines the maximum allowable process-model mismatch to guarantee stable control performance despite process uncertainty and disturbances. Proofs on the existence of a sequence of feasible approximations and control stability are given.Since plant approximations are built on-line based on actual measurements, the proposed controller can be interpreted as a multi-model nonlinear predictive control (MMPC). The performance of the MMPC strategy is illustrated by simulation experiments on a problem that involves reactant concentration control of a tubular reactor with recycle.     

Nonlinear model predictive control using parameter varying BP-ARX combination model

A novel back-propagation AutoRegressive with eXternal input (BP-ARX) combination model is constructed for model predictive control (MPC) of MIMO nonlinear systems, whose steady-state relation between inputs and outputs can be obtained. The BP neural network represents the steady-state relation, and the ARX model represents the linear dynamic relation between inputs and outputs of the nonlinear systems. The BP-ARX model is a global model and is identified offline, while the parameters of the ARX model are rescaled online according to BP neural network and operating data. Sequential quadratic programming is employed to solve the quadratic objective function online, and a shift coefficient is defined to constrain the effect time of the recursive least-squares algorithm. Thus, a parameter varying nonlinear MPC (PVNMPC) algorithm that responds quickly to large changes in system set-points and shows good dynamic performance when system outputs approach set-points is proposed. Simulation results in a multivariable stirred tank and a multivariable pH neutralisation process illustrate the applicability of the proposed method and comparisons of the control effect between PVNMPC and multivariable recursive generalised predictive controller are also performed.     

Distributed model predictive control of nonlinear systems subject to asynchronous and delayed measurements

In this work, we design distributed Lyapunov-based model predictive controllers for nonlinear systems that coordinate their actions and take asynchronous measurements and delays explicitly into account. Sufficient conditions under which the proposed distributed control designs guarantee that the state of the closed-loop system is ultimately bounded in a region that contains the origin are provided. The theoretical results are demonstrated through a chemical process example.     

A new approach to model reduction of nonlinear control systems using smooth orthogonal decomposition

A new approach to model order reduction of nonlinear control systems is aimed at developing persistent reduced order models (ROMs) that are robust to the changes in system's energy level. A multivariate analysis method called smooth orthogonal decomposition (SOD) is used to identify the dynamically relevant modal structures of the control system. The identified SOD subspaces are used to develop persistent ROMs. Performance of the resultant SOD‐based ROM is compared with proper orthogonal decomposition (POD)–based ROM by evaluating their robustness to the changes in system's energy level. Results show that SOD‐based ROMs are valid for a relatively wider range of the nonlinear control system's energy when compared with POD‐based models. In addition, the SOD‐based ROMs show considerably faster computations compared to the POD‐based ROMs of same order. For the considered dynamic system, SOD provides more effective reduction in dimension and complexity compared to POD.     

A contractive sliding-mode MPC algorithm for nonlinear discrete-time systems

This paper investigates a sliding-mode model predictive control (MPC) algorithm with auxiliary contractive sliding vector constraint for constrained nonlinear discrete-time systems. By adding contractive constraint into the optimization problem in regular sliding-mode MPC algorithm, the value of the sliding vector is decreased to zero asymptotically, which means that the system state is driven into a vicinity of sliding surface with a certain width. Then, the system state moves along the sliding surface to the equilibrium point within the vicinity. By applying the proposed algorithm, the stability of the closed-loop system is guaranteed. A numerical example of a continuous stirred tank reactor (CSTR) system is given to verify the feasibility and effectiveness of the proposed method.     

Distributed stochastic MPC for systems with parameter uncertainty and disturbances

A distributed stochastic model predictive control algorithm is proposed for multiple linear subsystems with both parameter uncertainty and stochastic disturbances, which are coupled via probabilistic constraints. To handle the probabilistic constraints, the system dynamics is first decomposed into a nominal part and an uncertain part. The uncertain part is further divided into 2 parts: the first one is constrained to lie in probabilistic tubes that are calculated offline through the use of the probabilistic information on disturbances, whereas the second one is constrained to lie in polytopic tubes whose volumes are optimized online and whose facets' orientations are determined offline. By permitting a single subsystem to optimize at each time step, the probabilistic constraints are then reduced into a set of linear deterministic constraints, and the online optimization problem is transformed into a convex optimization problem that can be performed efficiently. Furthermore, compared to a centralized control scheme, the distributed stochastic model predictive control algorithm only requires message transmissions when a subsystem is optimized, thereby offering greater flexibility in communication. By designing a tailored invariant terminal set for each subsystem, the proposed algorithm can achieve recursive feasibility, which, in turn, ensures closed‐loop stability of the entire system. A numerical example is given to illustrate the efficacy of the algorithm.     

Generalized sampled data feedback control of distributed parameter systems

In the sampled-data control literature there are necessary conditions and sufficient conditions for stabilizability of distributed parameter systems by generalized sampled-data control. For finite-dimensional systems the necessary conditions are also known to be sufficient. We show that this equivalence extends to the infinite-dimensional case if the underlying semigroup is analytic. However, for general systems, the necessary conditions are not sufficient, nor are the sufficient conditions necessary. We prove this by a single example with a free parameter – one choice of parameter shows that the necessary conditions are too weak, and another choice shows that the sufficient conditions are too strong.     

Input to state stability of min-max MPC controllers for nonlinear systems with bounded uncertainties

Min-max model predictive control (MPC) is one of the control techniques capable of robustly stabilize uncertain nonlinear systems subject to constraints. In this paper we extend existing results on robust stability of min-max MPC to the case of systems with uncertainties which depend on the state and the input and not necessarily decaying, i.e. state and input dependent bounded uncertainties. This allows us to consider both plant uncertainties and external disturbances in a less conservative way.It is shown that the input-to-state practical stability (ISpS) notion is suitable to analyze the stability of worst-case based controllers. Thus, we provide Lyapunov-like sufficient conditions for ISpS. Based on this, it is proved that if the terminal cost is an ISpS-Lyapunov function then the optimal cost is also an ISpS-Lyapunov function for the system controlled by the min-max MPC and hence, the controlled system is ISpS. Moreover, we show that if the system controlled by the terminal control law locally admits certain stability margin, then the system controlled by the min-max MPC retains the stability margin in the feasibility region.     

Obtaining initial parameter estimates for nonlinear systems using multicriteria associative memories

Parameter estimation problems for nonlinear systems are typically formulated as nonlinear optimization problems. For such problems, one has the usual difficulty that standard successive approximation schemes require good initial estimates for the parameter vector. This paper develops a simple multicriteria associative memory (MAM) procedure for obtaining useful initial parameter estimates for nonlinear systems. An easily calculated one-parameter family of associative memory matrices is developed; see Equation (25). Each memory matrix is efficient with respect to two criteria: accurate recovery of parameter-output training case associations; and small matrix norm to guard against noise arising from imprecise calculations and observations. For illustration, the MAM procedure is used to obtain initial parameter estimates for a well-known nonlinear economic model, the Solow-Swan growth model. Surprisingly accurate initial parameter estimates are obtained over broad ranges of the family of MAM memory matrices, even when observations are corrupted by i.i.d. or correlated noise.     

Well-posed problem of nonlinear singular distributed parameter systems and nonlinear GE-semigroup

According to the well-posed problem of nonlinear singular distributed parameter systems, first of all, the nonlinear GE-semigroup induced by a continuous （possibly nonlinear） operator is introduced in Banach space, which is a generalization of GE-semigroup （i.e., generalized operator semigroup）, and the properties of nonlinear GE-semigroup are discussed; and then the existence, uniqueness and constructive expression for the strong solution of nonlinear singular distributed parameter system are discussed by nonlinear GE-semigroup; at last, the exponential stability of nonlinear singular distributed parameter system is studied by using nonlinear GE-semigroup, functional analysis and operator theory in Banach space.     

Data-driven asymptotic stabilization for discrete-time nonlinear systems

In this paper, we propose a data-driven feedback controller design method based on Lyapunov approach, which can guarantee the asymptotic stability of the closed-loop and enlarge the estimate of domain of attraction (DOA) for the closed-loop. First, sufficient conditions for a feedback controller asymptotically stabilizing the discrete-time nonlinear plant are proposed. That is, if a feedback controller belongs to an open set consisting of pairs of control input and state, whose elements can make the difference of a control Lyapunov function (CLF) to be negative-definite, then the controller asymptotically stabilizes the plant. Then, for a given CLF candidate, an algorithm, to estimate the open set only using data, is proposed. With the estimate, it is checked whether the candidate is or is not a CLF. If it is, a feedback controller is designed just using data, which satisfies sufficient conditions mentioned above. Finally, the estimate of DOA for closed-loop is enlarged by finding an appropriate CLF from a CLF candidate set based on data. Because the controller is designed directly from data, complexity in building the model and modeling error are avoided.